In 1990, at the age of 33, Tsveiba (pictured) started his international career with the Soviet team, making 25 appearances for them through the summer of 1992 (the last seven of those appearances were with the Soviet Union's successor team, the Commonwealth of Independent States).
On 26 August 1992, after the dissolution of the CIS team, he made a lone appearance for Ukraine in a 1-2 friendly loss to Hungary. Then, just over five months later, he switched his allegiance to Russia. His first appearance for them came on 7 February 1997 in the opening match of the Carlsberg Cup, which Russia won over Yugoslavia on penalties, 1-1 (6-5). (Russia went on to win the tournament with a win over Switzerland in the final.)
He proceeded to make another seven appearances for Russia, bringing his career total to 34 before retiring from international play in 1997.
While players with caps from two different national teams are not uncommon, Tsveiba is one of only seven who have played for three different teams. Three of the others--Yury Nikiforov, Andrei Pyatnitskyi, and Sergei Mandreko--were former USSR players in similar situations to Tsveiba. The other three are Josef Bican (Austria, Czechoslovakia, and the Protectorate of Bohemia and Moravia), László Kubala (Czechoslovakia, Hungary, and Spain), and Karel Burkert (Bulgaria, Czechoslovakia, and Bohemia and Moravia).